Question: Solve for $x$ : $ 5|x + 5| + 8 = 4|x + 5| + 10 $
Explanation: Subtract $ {4|x + 5|} $ from both sides: $ \begin{eqnarray} 5|x + 5| + 8 &=& 4|x + 5| + 10 \\ \\ { - 4|x + 5|} && { - 4|x + 5|} \\ \\ 1|x + 5| + 8 &=& 10 \end{eqnarray} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} 1|x + 5| + 8 &=& 10 \\ \\ { - 8} &=& { - 8} \\ \\ 1|x + 5| &=& 2 \end{eqnarray} $ Simplify: $ |x + 5| = 2$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 5 = -2 $ or $ x + 5 = 2 $ Solve for the solution where $x + 5$ is negative: $ x + 5 = -2 $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& -2 \\ \\ {- 5} && {- 5} \\ \\ x &=& -2 - 5 \end{eqnarray} $ $ x = -7 $ Then calculate the solution where $x + 5$ is positive: $ x + 5 = 2 $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& 2 \\ \\ {- 5} && {- 5} \\ \\ x &=& 2 - 5 \end{eqnarray} $ $ x = -3 $ Thus, the correct answer is $x = -7 $ or $x = -3 $.